2024/10/14 by Ebeling, Alyssa, Hou, Xiang-dong, Rydell, Ashley +1
#11G25 #11T06 #11T71 #94D10 #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2410.10426
A function from \Bbb F2n to \Bbb F2n is said to be \em kth order sum-free if the sum of its values over each k-dimensional \Bbb F2-affine subspace of \Bbb F2n is nonzero. This notion was recently introduced by C. Carlet as, among other things, a generalization of APN functions. At the center of this new topic is a conjecture about the sum-freedom of the multiplicative inverse function f\rm inv(x)=x-1 (with 0-1 defined to be 0). It is known that f\rm inv is 2nd order (equivalently, (n-2)th order) sum-free if and only if n is odd, and it is conjectured that for 3≤ k≤ n-3, f\rm inv is never kth order sum-free. The conjecture has been confirmed for even n but remains open for odd n. In the present paper, we show that the conjecture holds under each of the following conditions: (1) n=13; (2) 3| n; (3) 5| n; (4) the smallest prime divisor l of n satisfies (l-1)(l+2)≤ (n+1)/2. We also determine the ``right'' q-ary generalization of the binary multiplicative inverse function f\rm inv in the context of sum-freedom. This q-ary generalization not only maintains most results for its binary version, but also exhibits some extraordinary phenomena that are not observed in the binary case.